F_spring ↔ F

Tier I — Equivalence PROVED

Equation A

EQ-HOOKE

F_spring

Object: rigid_body
Property: Force
Context: inertial_frame
Constraint: net_external

↔

Equation B

EQ-NEWTON-II

Object: rigid_body
Property: Force
Context: inertial_frame
Constraint: net_external

Sieve rationale

Identical tier1 key (object=rigid_body, property=Force, context=inertial_frame, constraint=net_external)

Physical constraint filter

tellegen pairing PASSED

Both variables are bond-graph efforts with equal pint dimensions ([mass] * [length] / [time] ** 2); this is a direct effort-identification across the coupling interface.

onsager reciprocity not applicable

Onsager reciprocity applies only to pairs of linear transport laws; at least one equation is not a linear PDE.

energy conservation PASSED

Newton+Hooke coupling admits the conserved energy E = (1/2) m v² + (1/2) k x²; dE/dt = 0 on trajectories of the composite SHO system.

Emergent properties

Conserved quantities

(1/2) * m * v**2 + (1/2) * k * x**2

Steady states

(x=0, v=0)

Linear stability

Class: center
Eigenvalues: -I*sqrt(k)/sqrt(m), I*sqrt(k)/sqrt(m)

Linear center: oscillations neither grow nor decay, amplitude is set by initial conditions. This is the simple harmonic oscillator, which is the canonical marginally-stable dynamical system.

Emergent Buckingham Π groups

Dimensionless combinations that mix parameters from both equations — only appear in the composite system.

F_spring^(-1) * k * x

F_spring^(-1) * F

F_spring^(-1) * a * m

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